Gopakumar–Vafa invariant

In theoretical physics, Rajesh Gopakumar and Cumrun Vafa introduced in a series of papers new topological invariants, called Gopakumar–Vafa invariants, that represent the number of BPS states on a Calabi–Yau 3-fold. They lead to the following generating function for the Gromov–Witten invariants on a Calabi–Yau 3-fold M:

${\displaystyle \sum _{g=0}^{\infty }~\sum _{\beta \in H_{2}(M,\mathbb {Z} )}{\text{GW}}(g,\beta )q^{\beta }\lambda ^{2g-2}=\sum _{g=0}^{\infty }~\sum _{k=1}^{\infty }~\sum _{\beta \in H_{2}(M,\mathbb {Z} )}{\text{BPS}}(g,\beta ){\frac {1}{k}}\left(2\sin \left({\frac {k\lambda }{2}}\right)\right)^{2g-2}q^{k\beta }}$ ,

where

• ${\displaystyle \beta }$ is the class of pseudoholomorphic curves with genus g,
• ${\displaystyle \lambda }$ is the topological string coupling,
• ${\displaystyle q^{\beta }=\exp(2\pi it_{\beta })}$ with ${\displaystyle t_{\beta }}$ the Kähler parameter of the curve class ${\displaystyle \beta }$,
• ${\displaystyle {\text{GW}}(g,\beta )}$ are the Gromov–Witten invariants of curve class ${\displaystyle \beta }$ at genus ${\displaystyle g}$,
• ${\displaystyle {\text{BPS}}(g,\beta )}$ are the number of BPS states (the Gopakumar–Vafa invariants) of curve class ${\displaystyle \beta }$ at genus ${\displaystyle g}$.